So when Killian said "I AM THE MANDARIN" it meant "I'm just the dude who hired another dude to disguise himself as another dude while I blow your house up"

I guess when Tony Stark said "I AM IRON MAN" it means "i'm just some dude who makes suits for the real tony stark"

okay lol

Yeah, cause if Stan Lee says "I am Spider-Man" that totally means he is, right? Oh wait, no. That's basically what this is. Killian created a fictional figure called "The Mandarin", but Killian does not represent any of this fictional character's beliefs or idealogies, nor is he in any way similar in style or in substance to said fictional character, so when he says "I am The Mandarin!" he is declaring himself as the figurehead of his organization, but he is not the actual character to which he claims the name of, he is a different one.